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Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady world Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady world Greg King University of Warwick (UK) Collaborators: • Murray Rudman (CSIRO) • George Rowlands (Warwick) • Thanasis Yannacopoulos (Aegean) • Katie Coughlin (LLNL) • Igor Mezic (UCSB)

To understand this lecture you need to know • • • Some fluid dynamics To understand this lecture you need to know • • • Some fluid dynamics Some Hamiltonian dynamics Something about phase space Poincare sections Need > 2 D phase space to get chaos Symmetry can reduce the dimensionality of phase space • Some knowledge of diffusion • A “friendly” applied mathematician !!

Phase Space Phase Space

Dynamical Systems and Phase Space Dynamical Systems and Phase Space

Classical Mechanics and Phase Space Hamiltonian Dissipative Classical Mechanics and Phase Space Hamiltonian Dissipative

Fluid Dynamics and Phase Space 2 D incompressible fluid 3 D incompressible fluid Phase Fluid Dynamics and Phase Space 2 D incompressible fluid 3 D incompressible fluid Phase Space No chaos here Symmetries -- can reduce phase space

Poincare Sections (Experimental – i. e. , light sheet) Poincare Sections (Experimental – i. e. , light sheet)

Eccentric Couette Flow Chaiken, Chevray, Tabor and Tan, Proc Roy Soc 1984 ? ? Eccentric Couette Flow Chaiken, Chevray, Tabor and Tan, Proc Roy Soc 1984 ? ? Illustrates “Significance” of KAM theory

Fountain et al, JFM 417, 265 -301 (2000) Stirring creates deformed vortex Fountain et al, JFM 417, 265 -301 (2000) Stirring creates deformed vortex

Fountain et al, JFM 417, 265 -301 (2000) Experiment (light sheet) Numerical Particle Tracking Fountain et al, JFM 417, 265 -301 (2000) Experiment (light sheet) Numerical Particle Tracking (“light sheet”)

Taylor-Couette Radius Ratio: = a/b Reynolds Number: b a Re = a (b-a)/ Taylor-Couette Radius Ratio: = a/b Reynolds Number: b a Re = a (b-a)/

Engineering Applications • • Chemical reactors Bioreactors Blood – Plasma separation etc Engineering Applications • • Chemical reactors Bioreactors Blood – Plasma separation etc

Taylor-Couette regime diagram (Andereck et al) Rein Reout Taylor-Couette regime diagram (Andereck et al) Rein Reout

Some Possible Flows Taylor vortices Twisted vortices Wavy vortices Spiral vortices Some Possible Flows Taylor vortices Twisted vortices Wavy vortices Spiral vortices

Taylor Vortex Flow TVF -– Centrifugal instability of circular Couette flow. – Periodic cellular Taylor Vortex Flow TVF -– Centrifugal instability of circular Couette flow. – Periodic cellular structure. – Three-dimensional, rotationally symmetric: u = u(r, z) Flat inflow and outflow boundaries are barriers to inter-vortex transport.

Rotational Symmetry 3 D 2 D Phase Space “Light Sheet” nested streamtubes /2 Z Rotational Symmetry 3 D 2 D Phase Space “Light Sheet” nested streamtubes /2 Z 0 inner cylinder Radius outer cylinder

Wavy Vortex Flow Taylor vortex flow wavy vortex flow Rec Wavy Vortex Flow Taylor vortex flow wavy vortex flow Rec

The Leaky Transport Barrier Wavy vortex flow is a deformation of rotationally symmetric Taylor The Leaky Transport Barrier Wavy vortex flow is a deformation of rotationally symmetric Taylor vortex flow. Increase Re Flow is steady in co-moving frame Poincare Sections Dividing stream surface breaks up => particles can migrate from vortex to vortex

Methods • • • Solve Navier-Stokes equations numerically to obtain wavy vortex flow. Finite Methods • • • Solve Navier-Stokes equations numerically to obtain wavy vortex flow. Finite differences (MAC method); Pseudo-spectral (P. S. Marcus) 2. Integrate particle path equations (20, 000 particles) in a frame rotating with the wave (4 th order Runge-Kutta).

Wavy Vortex Flow Poincare Section near onset of waves 6 vortices 1 2 3 Wavy Vortex Flow Poincare Section near onset of waves 6 vortices 1 2 3 4 5 6 inner cylinder outer cylinder

At larger Reynolds numbers (Rudman, Metcalfe, Graham: 1998) At larger Reynolds numbers (Rudman, Metcalfe, Graham: 1998)

Effective Diffusion Coefficient Characterize the migration of particles from vortex to vortex Rudman, AICh. Effective Diffusion Coefficient Characterize the migration of particles from vortex to vortex Rudman, AICh. E J 44 (1998) 1015 -26. Initialization: Uniformly distribute 20, 000 particles Taylor vortices (dimensionless) Wavy vortices

Size of mixing region Dz (dimensionless) Size of mixing region Dz (dimensionless)

Dz Dz

An Eulerian Approach Symmetry Measures Theoretical Fact A three dimensional phase space is necessary An Eulerian Approach Symmetry Measures Theoretical Fact A three dimensional phase space is necessary for chaotic trajectories. The Idea (Mezic): Deviation from certain continuous symmetries can be used to measure the local deviation from 2 D For Wavy Vortex Flow rotational symmetry and dynamical symmetry : • If either is zero, then flow is locally integrable, so as a diagnostic we consider the product

Dynamical Symmetry Steady incompressible Navier-Stokes equations in the form Equation of motion for B Dynamical Symmetry Steady incompressible Navier-Stokes equations in the form Equation of motion for B B is a constant of the motion if

155 Reynolds Number 162 324 486 648 155 Reynolds Number 162 324 486 648

155 Reynolds Number 162 324 486 648 155 Reynolds Number 162 324 486 648

155 Reynolds Number 162 324 486 648 Looks interesting, but correlation does not look 155 Reynolds Number 162 324 486 648 Looks interesting, but correlation does not look strong !

Averaged Symmetry Measures and partial averages Averaged Symmetry Measures and partial averages

Size of chaotic region Dz D Size of chaotic region Dz D

Serendipity ! King, Rudman, Rowlands and Yannacopoulos Physics of Fluids 2000 Serendipity ! King, Rudman, Rowlands and Yannacopoulos Physics of Fluids 2000

Effect of Radius Ratio (Mind the Gap) Dz/ Re/Rec Effect of Radius Ratio (Mind the Gap) Dz/ Re/Rec

Effect of Flow State : Axial wavelength m: Number of waves Re/Rec Effect of Flow State : Axial wavelength m: Number of waves Re/Rec

Effect of Flow State Dz Re/Rec Effect of Flow State Dz Re/Rec

Summary • Dz is highly correlated with < >< > • The correlation is Summary • Dz is highly correlated with < >< > • The correlation is not perfect. • The symmetry arguments are general • Yannacopoulos et al (Phys Fluids 14 2002) show that Melnikov function, M ~ < >. Is it good for anything else?

2 D Rotating Annulus u(r, z, t) Richard Keane’s results (see poster) Symmetry measure: 2 D Rotating Annulus u(r, z, t) Richard Keane’s results (see poster) Symmetry measure: FSLE Log(FSLE) Log(<|d /dt|>)

Prandtl-Batchelor Flows (Batchelor, JFM 1, 177 (1956) Steady Navier-Stokes equations in the form Integrating Prandtl-Batchelor Flows (Batchelor, JFM 1, 177 (1956) Steady Navier-Stokes equations in the form Integrating N-S equation around a closed streamline s yields

Break-up of Closed Streamlines Yannacopoulos et al, Phys Fluids 14 2002 (see also Mezic Break-up of Closed Streamlines Yannacopoulos et al, Phys Fluids 14 2002 (see also Mezic JFM 2001) Expand This is the Melnikov function